Which Hadamard Products of Generating Functions Are Known?

The Hadamard product, Schur product, or entrywise product of two generating functions is computed as follows:

The Hadamard Product, H(x), given two generating functions f(x) and g(x) where

$$ f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + dots + c_nx^n + dots $$
$$ g(x) = d_0 + d_1x + d_2x^2 + d_3x^3 + dots + d_nx^n + dots $$

is defined as

$$ H(x) = c_0d_0 + c_1d_1x + c_2d_2x^2 + c_3d_3x^3 + dots + c_nd_nx^n + dots $$

Simply put, it is the result of multiplying individual coefficients of two generating functions.

Results detailing certain types of functions are known. I would like to compile a list of results that are known.

For example, if we have two generating functions of the form $(1-x)^a$ and $(1-x)^b$, we obtain:
${}_2F_1[-a,-b;1;x]$ where ${}_2F_1$ represents a hypergeometric function of Gauss, according to

"Singularity Analysis, Hadamard Products, and Tree Recurrences", by Jim Fill, Philippe Flajolet, and Nevin Kapur In Journal of Computational and Applied Mathematics, volume 174 (February 2005), pages 271--313 (around page 289)

Again, which Hadamard products involving generating functions are known or solved? Additionally, solutions to the equations would be greatly appreciated.